Improved Algorithms for Economic Lot Size Problems

Aggarwal, Alok; Park, James K.

doi:10.1287/opre.41.3.549

Cited by 306 publications

(204 citation statements)

References 37 publications

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“…Corresponding to each candi-date, the first t period problem is a deterministic ULS with a given final inventory level, which is equivalent to another ULS without final inventory by updating the demand at time period t. Thus, the optimal solution can be obtained in O(n log n) time as studied by Aggarwal and Park (1993).…”

Section: Propositionmentioning

confidence: 99%

“…A simple dynamic programming algorithm based on this property runs in O(T 2 ) time, where T is the number of time periods; this was improved later to O(T log T ) or linear time for the case in which the costs exclude speculative motives (for details see Aggarwal and Park (1993), Federgruen and Tzur (1991), and Wagelmans et al (1992)). Polynomial time algorithms have also been developed for variants of the deterministic ULS.…”

Section: Introductionmentioning

confidence: 99%

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Polynomial-Time Algorithms for Stochastic Uncapacitated Lot-Sizing Problems

Guan

Miller

2008

Operations Research

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In 1958, Wagner and Whitin published a seminal paper on the deterministic uncapacitated lot-sizing problem, a fundamental model that is embedded in many practical production planning problems. In this paper we consider a basic version of this model in which demands (and other problem parameters) are stochastic: the stochastic uncapacitated lot-sizing problem (SULS). We define the production path property of an optimal solution for our model and use this property to develop a backward dynamic programming recursion. This approach allows us to show that the value function is piecewise linear and right continuous. We then use these results to show that a full characterization of the optimal value function can be obtained by a dynamic programming algorithm in polynomial time for the case that each non-leaf node contains at least two children.Moreover, we show that our approach leads to a polynomial time algorithm to obtain an optimal solution to any instance of SULS, regardless of the structure of the scenario tree. We also show that the value function for the problem without setup costs is continuous, piecewise linear, and convex, and therefore an even more efficient dynamic programming algorithm can be developed for this special case.

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Section: Propositionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Polynomial-Time Algorithms for Stochastic Uncapacitated Lot-Sizing Problems

Guan

Miller

2008

Operations Research

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“…In the last decade, three important papers, Aggarwal andPark (1993), Federgruen andTzur (1991), and Wagelmans et al (1992), improved the time complexity for obtaining an optimal solution from O(T 2 ) to O(TlogT) for general problems, and to O(T) for problems with a special cost structure. For interesting generalizations of Wagner and Whitin's model, see Bitran et al (1984), Chen et al (1994), Lee (1995), Chung andLin (1988), Florian et al (1980), Wagelmans (1996, 1997), Lee (1989), Lee and Denardo (1986), Maes and Van Wassenhove (1985), Swoveland (1975), andZangwill (1966).…”

Section: Problem Motivations and Related Literaturementioning

confidence: 99%

“…Remark 1: Recently, Federgruen and Tzur (1991), Wagelmans et al (1992), and Aggarwal and Park (1993) have developed O(T) algorithms to solve the classical dynamic lot-sizing problems for which n ≤ T. One of the key ideas in those papers is to transform the mathematical formulation of the classical dynamic lot-sizing model to an equivalent formulation without holding costs. Unfortunately, for the more general problem with demand time window considerations, equation (4) seems to make such a transformation difficult.…”

Section: Recall That the Computational Complexity Of Calculating B(smentioning

confidence: 99%

A Dynamic Lot-Sizing Model with Demand Time Windows

2001

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Abstract:One of the basic assumptions of the classical dynamic lot-sizing model is that the aggregate demand of a given period must be satisfied in that period. Under this assumption, if backlogging is not allowed then the demand of a given period cannot be delivered earlier or later than the period. If backlogging is allowed, the demand of a given period cannot be delivered earlier than the period, but can be delivered later at the expense of a backordering cost. Like most mathematical models, the classical dynamic lot-sizing model is a simplified paraphrase of what might actually happen in real life. In most real life applications, the customer offers a grace period -we call it a demand time window -during which a particular demand can be satisfied with no penalty. That is, in association with each demand, the customer specifies an earliest and a latest delivery time. The time interval characterized by the earliest and latest delivery dates of a demand represents the corresponding time window. This paper studies the dynamic lot-sizing problem with demand time windows and provides polynomial time algorithms for computing its solution. If shortages are not allowed, the complexity of the proposed algorithm is O(T 2 ). When backlogging is allowed, the complexity of the proposed algorithm is O(T 3 ).2

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“…Florian et al (1980) have in fact shown that even the single-item case N = 1 is NP-complete, as opposed to the uncapacitated version that, for a planning horizon of T periods, is solvable in O T log T time (see Federgruen and Tzur 1991, Wagelmans et al 1992, and Aggarwal and Park 1993, and in O T time under some mild assumptions on the data. The difficulty arises in part because under capacity restrictions, it may no longer be optimal to place an order at the last possible time; in other words, it is not possible to confine oneself to so-called zero-inventory ordering policies.…”

Section: Introductionmentioning

confidence: 99%

Progressive Interval Heuristics for Multi-Item Capacitated Lot-Sizing Problems

Federgruen

Meissner

Tzur

2007

Operations Research

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We consider a family of N items that are produced in, or obtained from, the same production facility. Demands are deterministic for each item and each period within a given horizon of T periods. If in a given period an order is placed, setup costs are incurred. The aggregate order size is constrained by a capacity limit. The objective is to find a lot-sizing strategy that satisfies the demands for all items over the entire horizon without backlogging, and that minimizes the sum of inventory-carrying costs, fixed-order costs, and variable-order costs. All demands, cost parameters, and capacity limits may be time dependent. In the basic joint setup cost (JS) model, the setup cost of an order does not depend on the composition of the order. The joint and item-dependent setup cost (JIS) model allows for item-dependent setup costs in addition to the joint setup costs.We develop and analyze a class of so-called progressive interval heuristics. A progessive interval heuristic solves a JS or JIS problem over a progressively larger time interval, always starting with period 1, but fixing the setup variables of a progressively larger number of periods at their optimal values in earlier iterations. Different variants in this class of heuristics allow for different degrees of flexibility in adjusting continuous variables determined in earlier iterations of the algorithm.For the JS-model and the two basic implementations of the progressive interval heuristics, we show under some mild parameter conditions that the heuristics can be designed to be -optimal for any desired value of > 0 with a running time that is polynomially bounded in the size of the problem. They can also be designed to be simultaneously asymptotically optimal and polynomially bounded.A numerical study covering both the JS and JIS models shows that a progressive interval heuristic generates close-tooptimal solutions with modest computational effort and that it can be effectively used to solve large-scale problems.

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Improved Algorithms for Economic Lot Size Problems

Cited by 306 publications

References 37 publications

Polynomial-Time Algorithms for Stochastic Uncapacitated Lot-Sizing Problems

Polynomial-Time Algorithms for Stochastic Uncapacitated Lot-Sizing Problems

A Dynamic Lot-Sizing Model with Demand Time Windows

Progressive Interval Heuristics for Multi-Item Capacitated Lot-Sizing Problems

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